$證明:∵四邊形???ABCD???為矩形$
$??? ∴∠B=∠C=90°,??????AB=CD???$
$??? ∵EF⊥ED ???$
$??? ∴∠BEF+∠CED=∠FED=90°???$
$??? ∵∠BEF+∠BFE=90°???$
$??? ∴∠BFE=∠CED???$
$ 在???△BFE???和???△CED???中$
$??? \begin{cases}∠B=∠C\\∠BFE=∠CED\\EF=ED\end{cases}???$
$??? ∴△BFE≌△CED(\mathrm {AAS})???$
$??? ∴BE=CD???$
$??? ∴AB=BE,???$
$??? ∴∠BAE=45°???$
$??? ∴AE???平分???∠BAD???$